The speckle intensities of both strong scatterers and weak scatterers are modelled as the superposition of N sinusoidal waves whose amplitudes are taken to be correlated random variables in order to account for multipath correlations. Explicit expressions are obtained for the first three moments of the speckle intensity when the correlated amplitudes are gaussian-distributed about a non-zero mean. These moments approach their large-Nasymptotic values independently of the amplitude correlation function. The contrast and skewness parameter are also evaluated, and simplified formulae obtained when N is large. As the number of multipaths become very large, the speckle intensity PDFs tend to limiting functional forms by reason of the central limit theorem; an investigation was made to determine the influence of amplitude correlations on the parameters entering into these limiting functional forms. The implication of these very large-Nasymptotic results is that amplitude correlations are probably not significant in so far as altering the basic structure of the speckle intensity PDF. However, whenNis not large, the amplitude correlations play a significant role.